A new cosmological model is proposed, which is based on two important intuitions of Albert Einstein. The model is admittedly conjectural, since it is based on gravitational repulsion between matter and antimatter, which is theoretically foreseen by some theorists, but denied by the first experiment. The values currently considered correct for some important system parameters (mass budget, Hubble constant, universe age) are assumed. Universe dimensions, expansion velocity and total mass are computed. The highly questioned inflation hypothesis is abandoned, and the matter-antimatter unbalance strongly reduced. As a consequence, the dark side of the cosmos assumes new, much smaller, dimensions. Another result of major importance is the time allowed for the formation of the first structures, which is much larger than predicted by traditional Big Bang models. A problem of horizon does not seem to exist: all parts of the universe are visible by every observer; this was true also in the past, with the exception of a relatively short initial period, and will be true also in the future. As concerns CMBR anisotropy, the structure seed dimension given by the model looks in reasonable agreement with the experimental data provided by the scientific satellites.The first part of the paper shortly reviews the current situation of cosmology. In the second part the new model is discussed and evaluated from a quantitative viewpoint. The third part describes the novel spacetime map, including the derivation from measured redshift of comoving distance and time after Big Bang, and the discussion of the horizon problem.